The Exponential Localization and Structure of the Spectrum for 1d Quasi-Periodic Discrete SCHRÖDINGER Operators

We discuss main mechanisms of the exponential localization of the eigenfunctions for one-dimensional quasi-periodic Schrödinger operators with the potential of the form V(α + nω), where V(α) is a non-degenerate C²-function on the d-dimensional torus, and ω ∈ &R;^d is a typical vector with rationally incommensurate components. The exponential localization is proved so far for d ≤ 2. We emphasize the different nature of the support of the spectral measure for d = 1 and for d > 1.